<<

2019 Fall

Introduction to and

09. 19. 2019 Eun Soo Park

Office: 33‐313 Telephone: 880‐7221 Email: [email protected] Office hours: by appointment 1 Contents for previous class CHAPTER 3: Fundamentals of

I. Structures - , Unit Cells,

II. Crystallographic Points, Directions, and Planes - Point coordinates, Crystallographic directions, Crystallographic planes

III. Crystalline and Noncrystalline Materials - Single , Polycrystalline materials, , Noncrystalline

2 Stacking of in Finding stable position

2 1

• Minimize configuration – Related to the bonding nature

Lattice : 결정 공간상에서 점들의 규칙적인 기하학적 배열 – 3D point array in space, such that each point has identical surroundings. These points may or may not coincide with positions. – Simplest case : each atom  its center of gravity  point or space lattice  pure mathematical concept

example: sodium (Na) ; body centered cubic

Hard‐sphere unit cell Reduced sphere unit cell Aggregate of many atoms I. Crystal structure : Unit cell

14 Bravais Lattice - Only 14 different types of unit cells are required to describe all lattices using symmetry

rhombohedral cubic hexagonal tetragonal orthorhombic monoclinic triclinic (trigonal)

P

I

F

C

5 II. Crystallographic points, directions and planes

Chapter 3.5 Point coordinates • position: fractional multiples of the unit cell edge lengths – ex) P: q,r,s

cubic unit cell 6 Crystallographic Directions • a line between two points or a vector • [uvw] square bracket, smallest integer • families of directions: angle bracket cubic

7 Chapter 3.7 Crystallographic Planes Lattice plane (Miller indices) z p m00, 0n0, 00p: define lattice plane  m, n, ∞ : no intercepts with axes c Intercepts @ 213   n y (mnp) a b Reciprocals ½ 1 1/3 Miller indicies 3 6 2 m (362) plane x Plane (hkl) Family of planes {hkl}

Miller indicies ; defined as the (362) smallest integral multiples of the reciprocals of the plane intercepts on the axes 8 Miller-Bravais vs. Miller index system Directions in Hexagonal system

Conversion of 4 index system (Miller-Bravais) to 3 index (Miller)         t  u'a  v'a  w'c  ua  va  ta  wc Ex. M [100] 1 2 1 2 3 u=(1/3)(2*1-0)=2/3 v=(1/3)(2*0-1)=-1/3 Miller-Bravais to Miller Miller to Miller-Bravais w=0 4 to 3 axis 3axis to 4 axis system => 1/3[2 -1 -1 0] 1 u' u  t  2u  v u  (2u'v') 3 Ex. M-B [1 0 -1 0] u’=2*1+0=2 v' v  t  2v  u 1 v  (2v'u') v’=2*0+1=1 w' w 3 w’=0 => [2 1 0] w  w' 9 Contents for today’s class CHAPTER 3: Fundamentals of Crystallography III. Crystalline ↔ Noncrystalline Materials - Single crystals, Polycrystalline materials, Anisotropy - - Noncrystalline solids :

CHAPTER 4: The Structure of Crystalline Solids

10 III. Crystalline ↔ Noncrystalline Materials

CRYSTALS AS BUILDING BLOCKS • Some engineering applications require single crystals: -- single crystals --turbine blades Natural and artificial Fig. 8.30(c), Callister 6e. (Fig. 8.30(c) courtesy of Pratt and Whitney).

(Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.)

• Crystal properties reveal features of atomic structure. --Ex: Certain crystal planes in quartz fracture more easily than others. (Courtesy P.M. Anderson) 11 Solidification: Liquid Solid

4 Fold Anisotropic Surface Energy/2 Fold , Many Seeds 12 Grain Boundaries

13 Single vs Polycrystals

• Single Crystals E (diagonal) = 273 GPa Data from Table 3.3, -Properties vary with Callister 7e. (Source of data is R.W. direction: anisotropic. Hertzberg, Deformation and Fracture of -Example: the modulus Engineering Materials, 3rd ed., John Wiley and Sons, of (E) in BCC : 1989.)

E (edge) = 125 GPa • Polycrystals -Properties may/may not 200 m Adapted from Fig. 4.14(b), Callister 7e. vary with direction. (Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, -If grains are randomly the National Bureau of Standards, Washington, oriented: isotropic. DC [now the National Institute of Standards and (Epoly iron = 210 GPa) Technology, Gaithersburg, -If grains are textured, MD].) anisotropic. 14 Polycrystals Anisotropic • Most engineering materials are polycrystals.

Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

• Nb-Hf-W plate with an weld. Isotropic • Each "grain" is a . • If grains are randomly oriented, overall component properties are not directional. • Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). 15 Polymorphism

• Two or more distinct crystal structures for the same (allotropy/polymorphism) iron system liquid , -Ti 1538ºC -Fe BCC diamond, 1394ºC FCC -Fe 912ºC BCC -Fe

16 DEMO: HEATING AND COOLING OF AN IRON WIRE

The same atoms can • Demonstrates "polymorphism" have more than one crystal structure. , C Liquid 1536 BCC Stable 1391

longer up FCC Stable shorter! 914 BCC Stable longer! cool down Tc 768 magnet falls off shorter

17 Non-crystalline Materials

Amorphous materials a wide diversity of materials can be rendered amorphous indeed almost all materials can. - , ,

ex) amorphous metallic (1960)

- glassy/non-crystalline material cf) amorphous vs Amorphous - random atomic structure (short range order) - showing . Glass

- retain liquid structure - rapid solidification from liquid state

18 Obsidian is a naturally occurring volcanic glass formed as an extrusive igneous rock. It is produced when felsic lava extruded from a volcano cools rapidly with minimum . Obsidian is commonly found within the margins of rhyolitic lava flows known as obsidian flows, where the chemical composition (high silica content) induces a high and polymerization degree of the lava. The inhibition of atomic through this highly viscous and polymerized lava explains the lack of crystal growth. Because of this lack of crystal structure, abcde obsidian blade edges can reach almost molecular thinness, leading to its ancient use as projectile points and blades, and its modern use as surgical scalpel blades. Glass formation: stabilizing the liquid

First metallic glass (Au80Si20) produced by splat at Caltech by Pol Duwez in 1957.

1500 1414 -RapidRelative splat decrease quenching of 1300 Liquid mix meltingliquid metal temperature droplet Tm 1064.4 laser 1100 T mix T trigger T*  m m  0.2 900 mix deep eutectic Tm metal anvil 700 ΔT* = 0.679 in most of glass forming alloys

500 mix i 363 (where, T m   x iT m , 18.6 metal 300 xi = mole fraction,piston Tm i 100 Tm = melting point) 01020 30 40 50 60 70 80 90 100 Au Si t = 20 μm w = 2 cm W. Klement, R.H. Willens, P. Duwez, Nature 1960; 187: 869. by I.W. Donald et al, J. Non-Cryst. Solids, l1978;30:77. = 3 cm 20 Bulk glass formation in the Pd-/Ni-/Cu-/Zr- system

21 Recent BMGs with critical size ≥ 10 mm

22 Strength vs. Elastic Limit for Several Classes of Materials 2500

2000

1500 Titanium Metallic glass alloys

1000

500 Woods Silica 0 0 1 2 3 Elastic Limit (%) : Metallic Offer a Unique Combination of High Strength and High Elastic Limit

23 Processing as efficiently as : net‐shape forming!

Seamaster Planet Ocean Liquidmetal® Limited Edition Superior thermo‐ formability : possible to fabricate complex structure without joints Multistep processing can be solved by simple casting Ideal for small expensive IT equipment manufacturing

24 What are Quasicrystals?

25 Crystals can only exhibit certain symmetries

In crystals, atoms or atomic clusters repeat periodically, analogous to a tesselation in 2D constructed from a single type of tile.

Try tiling the plane with identical units … only certain symmetries are possible

A tessellation or tiling of the plane is 26 a collection of plane figures that fills the plane with no overlaps and no gaps. Symmetry axes compatible with periodicity

According to the well-known theorems of crystallography, only certain symmetries are allowed: the symmetry of a square, rectangle, parallelogram, triangle or hexagon, but not others, such as pentagons.

27 Crystals can only exhibit certain symmetries

Crystals can only exhibit these same rotational symmetries*

..and the symmetries determine many of their physical properties and applications

* In 3D, there can be different rotational symmetries along different axes, but they are restricted to the same set (2-, 3-, 4-, and 6-fold)

28 Quasicrystals (Impossible Crystals)

were first discovered in the laboratory by

Daniel Shechtman, Ilan Blech, Denis Gratias and John Cahn in a beautiful study of an alloy of Al and Mn

29 D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984)

1 m

Al6Mn

30 Their surprising claim: “Diffracts like a crystal . . . But with a symmetry strictly forbidden for crystals”

Al6Mn

31 QUASICRYSTALS Similar to crystals, BUT… • Orderly arrangement . . . But QUASIPERIODIC instead of PERIODIC

• Rotational Symmetry . . . But with FORBIDDEN symmetry

• Structure can be reduced to a finite number of repeating units

D. Levine and P.J. Steinhardt (1984) Discovery of a Natural L Bindi, P. Steinhardt, N. Yao and P. Lu Science 324, 1306 (2009)

LEFT: Fig. 1 (A) The original -bearing sample used in the study. The lighter-colored material on the exterior containsa

of , , and . The dark material consists predominantly of khatyrkite (CuAl2) and (CuAl) but also includes granules, like the one in (B), with composition Al63Cu24Fe13. The diffraction patterns in Fig. 4 were obtained from the thin region of this granule indicated by the red dashed circle, an area 0.1 μm across. (C) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodically ordered, fivefold symmetric, real space pattern characteristic of quasicrystals. 33 RIGHT: Diffraction patterns obtained from natural quasicrystal grain 2011 Nobel Prize in : Quasicrystal

A new ordered phase showing the apparent fivefold symmetry was observed by Sastry et al. [Mater. Res. Bull. 13: 1065-1070] in 1978 in a rapidly solidified Al-Pd alloy, but was interpreted to arise from a Consisting of a series of fine twins. This was later shown to be a two-dimensional (or decagonal) qasicrystal. Quasicrystals Crystal with 5 fold symmetry Mathematically impossible but exist

1984 Al86Mn14 alloy : rapidly solidified ribbon_Shectman et al. : materials whose structure cannot be understood within classical crystallography rules. “Quasiperiodic lattices”, with long-range order but without periodic translations in three dimensions

• long range order: quasiperiodic • no 3-D translational symmetry • sharp diffraction patterns http://www.youtube.com/watch?v=k_VSpBI5EGM 35 Atomic arrangement in the solid state  Solid materials are classified according to the regularity with which atoms and are arranged with respect to one another.

 So, how are they arranged ?

(a) periodically – having long range order in 3-D Crystal

(b) quasi-periodically Quasicrystal

(c) randomly – having short range order with the characteristics of bonding type but losing the long range order

 Crystal: Perfection → Imperfection Amorphous

36 CHAPTER 4: The Structure of Crystalline Solids

I. METALLIC CRYSTALS

• tend to be densely packed. • have several reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy. • have the simplest crystal structures. We will look at three such structures...

37 (1) SIMPLE CUBIC STRUCTURE (SC)

• Rare due to poor packing (only 84Polonium has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors)

(Courtesy P.M. Anderson)

In chemistry and crystallography, the coordination number (CN) of a central atom in a or crystal is the number of its nearest neighbor.

In chemistry the emphasis is on bonding structures in or ions and the CN of an atom is determined by simply counting the other of atoms to which it is bonded (by either single or multiple bonds).

The solid-state structure of crystals often have less clearly defined bonds, so a simpler model is used, in which the atoms are represented by touching spheres. In this model the CN of an atom is the number of other atoms which it touches. For an atom in the interior of a crystal lattice, the number of atoms touching the given atom is the bulk coordination number, for an atom at a surface of a crystal this is the surface coordination number.

39 Coordination Number

40 Coordination Number

41 ATOMIC PACKING FACTOR : simple cubic Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres

• APF for a simple cubic structure = 0.52 volume atoms 4 atom a unit cell 1 (0.5a)3 3 R=0.5a APF = a3 volume close-packed directions unit cell contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.19, 42 Callister 6e. (2) Body Centered Cubic Structure (BCC)

• Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Fe (), Tantalum, Molybdenum • Coordination # = 8

Adapted from Fig. 3.2, Callister 7e. 2 atoms/unit cell: 1 center + 8 corners x 1/8 (Courtesy P.M. Anderson) 43 44 Body Centered Cubic (BCC) • Close packed directions are cube diagonals <111> 1/2 • 2 atoms / unit cell (1 + 8 x 1/8) aBCC = 4r/3 • Highest density plane {110} Coordination # = 8 • Cr, Fe, W, etc.

• Coordination # = 8

r aBCC 45 ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68

Close-packed directions: length = 4R = 3 a 3 a

Unit cell contains: a 1 + 8 x 1/8 = 2 atoms/unit cell 2 a R a Close-packed directions <111> atoms volume 4 3 length = 4R = 3 a unit cell 2 ( 3a/4) 3 atom APF = Highest density planes volume a3 {110} unit cell Adapted from 46 Callister 6e. {110} Family

47 {110} Family

48 ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68

Close-packed directions: length = 4R = 3 a 3 a

Unit cell contains: a 1 + 8 x 1/8 = 2 atoms/unit cell 2 a R a Close-packed directions <111> atoms volume 4 3 length = 4R = 3 a unit cell 2 ( 3a/4) 3 atom APF = Highest density planes volume a3 {110} unit cell Adapted from 49 Callister 6e. Planar Density of (100) Iron At T < 912C iron has the BCC structure.

2D repeat unit

4 3 (100) a  R 3

Adapted from Fig. 3.2(c), Callister 7e. Radius of iron R = 0.1241 nm atoms 2D repeat unit 1 1 atoms atoms Planar Density = = = 12.1 = 1.2 x 1019 2 2 2 2 area a 4 3 nm m R 2D repeat unit 3 50 Planar Density of (111) Iron

(111) plane 1 atom in plane/ unit surface cell

2 a atoms in plane atoms above plane atoms below plane

3 h  a 2 2 4 3 16 3 area  2 ah  3 a 2  3 R  R2 3 3 atoms 2D repeat unit 1 atoms atoms Planar Density = = 7.0 = 0.70 x 1019 2 2 area 16 3 nm m R 2 2D repeat unit 3 51 Metallic crystal system

• Stacking sequence

A A A A A B BBB C C C C A A A A A B BBB C C C C A A A A A B BBB C C C C A A A A A B BBB C C C C A A A A A

52 Metallic crystal system

•A– B – C – A stacking sequence  FCC

A A A A A BB BBBB BB B CC C C C A A A A C A C A BB BBBB BB C CC CC C C B A A A C A C A BB BBBB BB A CC CC CC C A A A A A C BB BBB B BB B C C C CC A A A A A A

53 Metallic crystal system

•A – B – A – B stacking sequence  HCP

AA AA AA AA AA A BB BBBB BB C C C C B AA AA AA AA AA BB BBBB BB A C C C C AA AA AA AA AA B BB BBBB BB C C C C A AA AA AA AA AA BB BBB B BB B C C C C AA AA AA AA AA A (3) Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag • Coordination # = 12

Adapted from Fig. 3.1, Callister 7e.

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

(Courtesy P.M. Anderson) 55 ATOMIC PACKING FACTOR: FCC • APF for a face-centered cubic structure = 0.74 Close-packed directions: <110> length = 4R {111} = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a Adapted from Fig. 3.1(a), Callister 6e. atoms 4 volume unit cell 4 ( 2a/4)3 3 atom APF = volume a3 unit cell 56 Two closely packed stackings

A position B position C position

FCC A-B-C-A-B-C-….. HCP A-B-A-B-A-…..

FCC : Al, Ni, Cu, Ag, Ir, Pt, Au HCP : Be, Mg, Sc, Ti, Co, Zn, Y, Zr, Se, Te, Ru HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)

• ABAB... Stacking Sequence • 3D Projection • 2D Projection

A sites Top layer

B sites Middle layer

A sites Bottom layer

Adapted from Fig. 3.3, Callister 6e. • Coordination # = 12 Close-packed direction : <1120> • APF = 0.74 plane : (0001)

58 Contents for today’s class I. Atomic arrangement in the solid state

 Solid materials are classified according to the regularity with which atoms and ions are arranged with respect to one another.

 So, how are they arranged ?

(a) periodically – having long range order in 3-D Crystal

(b) quasi-periodically Quasicrystal

(c) randomly – having short range order with the characteristics of bonding type but losing the long range order

 Crystal: Perfection → Imperfection Amorphous 59 Contents for today’s class II. Metallic crystal system